Download presentation

Presentation is loading. Please wait.

Published byEustace Cunningham Modified over 5 years ago

1
Midwestern State University Minimum Spanning Trees Definition of MST Generic MST algorithm Kruskal's algorithm Prim's algorithm 1

2
2 Definition of MST Let G=(V,E) be a connected, undirected graph. For each edge (u,v) in E, we have a weight w(u,v) specifying the cost (length of edge) to connect u and v. We wish to find a (acyclic) subset T of E that connects all of the vertices in V and whose total weight is minimized. Since the total weight is minimized, the subset T must be acyclic (no circuit). Thus, T is a tree. We call it a spanning tree. The problem of determining the tree T is called the minimum-spanning-tree problem.

3
3 Application of MST: an example In the design of electronic circuitry, it is often necessary to make a set of pins electrically equivalent by wiring them together. Running cable TV to a set of houses. What’s the least amount of cable needed to still connect all the houses?

4
4 Here is an example of a connected graph and its minimum spanning tree: a b h cd e fg i 4 87 9 10 14 4 2 2 6 1 7 11 8 Notice that the tree is not unique: replacing (b,c) with (a,h) yields another spanning tree with the same minimum weight.

5
5 Growing a MST Set A is always a subset of some minimum spanning tree.. An edge (u,v) is a safe edge for A if by adding (u,v) to the subset A, we still have a minimum spanning tree. GENERIC_MST(G,w) 1A:={} 2while A does not form a spanning tree do 3find an edge (u,v) that is safe for A 4A:=A ∪ {(u,v)} 5return A

6
6 How to find a safe edge We need some definitions and a theorem. A cut (S,V-S) of an undirected graph G=(V,E) is a partition of V. An edge crosses the cut (S,V-S) if one of its endpoints is in S and the other is in V-S. An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

7
7 V-S↓ a b h cd e fg i 4 87 9 10 14 4 2 2 6 1 7 11 8 S↑S↑ ↑ S ↓ V-S This figure shows a cut (S,V-S) of the graph. The edge (d,c) is the unique light edge crossing the cut.

8
8 The algorithms of Kruskal and Prim The two algorithms are elaborations of the generic algorithm. They each use a specific rule to determine a safe edge in the GENERIC_MST. In Kruskal's algorithm, –The set A is a forest. –The safe edge added to A is always a least-weight edge in the graph that connects two distinct components. In Prim's algorithm, –The set A forms a single tree. –The safe edge added to A is always a least-weight edge connecting the tree to a vertex not in the tree.

9
9 Kruskal's algorithm (simple) (Sort the edges in an increasing order) A:={} while (E is not empty do) { take an edge (u, v) that is shortest in E and delete it from E If (u and v are in different components)then add (u, v) to A } Note: each time a shortest edge in E is considered.

10
10 Kruskal's algorithm 1 function Kruskal(G = : graph; length: A → R+): set of edges 2 Define an elementary cluster C(v) ← {v}. 3 Initialize a priority queue Q to contain all edges in G, using the weights as keys. 4 Define a forest T ← Ø //T will ultimately contain the edges of the MST 5 // n is total number of vertices 6 while T has fewer than n-1 edges do 7 // edge u,v is the minimum weighted route from u to v 8 (u,v) ← Q.removeMin() 9 // prevent cycles in T. add u,v only if T does not already contain a path // between u and v. 10 // the vertices has been added to the tree. 11 Let C(v) be the cluster containing v, and let C(u) be the cluster containing u. 13 if C(v) ≠ C(u) then 14 Add edge (v,u) to T. 15 Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u). 16 return tree T

11
11 http://Wikipedia/kruskals This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted. Kruskal's algorithm

12
12 http://Wikipedia/kruskals AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted. Kruskal's algorithm

13
13 http://Wikipedia/kruskals CE is now the shortest arc that does not form a cycle, with length 5, so it is highlighted as the second arc. Kruskal's algorithm

14
14 http://Wikipedia/kruskals The next arc, DF with length 6, is highlighted using much the same method. Kruskal's algorithm

15
15 http://Wikipedia/kruskals The next-shortest arcs are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The arc BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen. Kruskal's algorithm

16
16 http://Wikipedia/kruskals The process continues to highlight the next-smallest arc, BE with length 7. Many more arcs are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD. Kruskal's algorithm

17
17 http://Wikipedia/kruskals Finally, the process finishes with the arc EG of length 9, and the minimum spanning tree is found. Kruskal's algorithm

18
18 Prim's algorithm (simple) MST_PRIM(G,w,r){ A={} S:={r} (r is an arbitrary node in V) Q=V-{r}; while Q is not empty do { take an edge (u, v) such that u S and v Q (v S ) and (u,v) is the shortest edge add (u, v) to A, add v to S and delete v from Q }

19
19 Prim's algorithm for each vertex in graph set min_distance of vertex to ∞ set parent of vertex to null set minimum_adjacency_list of vertex to empty list set is_in_Q of vertex to true set min_distance of initial vertex to zero add to minimum-heap Q all vertices in graph, keyed by min_distance Initialization inputs: A graph, a function returning edge weights weight-function, and an initial vertex Initial placement of all vertices in the 'not yet seen' set, set initial vertex to be added to the tree, and place all vertices in a min-heap to allow for removal of the min distance from the minimum graph. Wikipedia

20
20 Prim's algorithm while latest_addition = remove minimum in Q set is_in_Q of latest_addition to false add latest_addition to (minimum_adjacency_list of (parent of latest_addition)) add (parent of latest_addition) to (minimum_adjacency_list of latest_addition) for each adjacent of latest_addition if ((is_in_Q of adjacent){ and (weight-function(latest_addition, adjacent) < min_distance of adjacent)) set parent of adjacent to latest_addition set min_distance of adjacent to weight-function(latest_addition, adjacent) } update adjacent in Q, order by min_distance Wikipedia The while loop will fail when remove minimum returns null. The adjacency list is set to allow a directional graph to be returned. time complexity: V for loop, log(V) for the remove function time complexity: E/V, the average number of vertices time complexity: log(V), the height of the heap

21
21 Grow the minimum spanning tree from the root vertex r. Q is a priority queue, holding all vertices that are not in the tree now. key[v] is the minimum weight of any edge connecting v to a vertex in the tree. parent[v] names the parent of v in the tree. When the algorithm terminates, Q is empty; the minimum spanning tree A for G is thus A={(v,parent[v]):v ∈ V-{r}}. Running time: O(||E||lg |V|). Prim's algorithm

22
22 Prim's algorithm

23
23 Prim's algorithm

24
24 Prim's algorithm

25
25 Prim's algorithm

26
26 Prim's algorithm

27
27 Prim's algorithm

28
28 Prim's algorithm

29
29 Prim's algorithm

30
30 Prim's algorithm

31
31 Prim's algorithm

32
32 Prim's algorithm

33
33 Prim's algorithm

34
34 Prim's algorithm

35
35 Prim's algorithm

36
36 Prim's algorithm

37
37 Prim's algorithm

38
38 Prim's algorithm

39
39 Prim's algorithm

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google